Wednesday, July 14, 2010

Solution of Quantum Mechanics Problems

Let’s look at the solutions of previous problems in quantum mechanics:

1) An electron is in the n= 2 state of hydrogen. Find the possible values of: L, L(z) and theta.

If the angular momentum quantum number l = 2

Then: L = [l(l + 1)]^1/2 (h/2pi) = [2(2 +1)]^1/2 (h/2pi)

= [6]^1/2 (h/2pi)

and a set of allowed projections for L are obtained (see diagram ) as well as orientations. Of particular interest are the allowed values for the vertical component of the orbital angular momentum vector, which we designate: L(z). (L(z) is the projection of L along the z-axis and has discrete values, e.g. 0, h/2pi, h/pi, - h/2pi, - h/pi for the case given).[See Diagram]

L(z) = m_l(h/2pi)

Recall that m_l can range from -l to l, so for l = 2, we have:

m_l = +2, +1, 0, -1, -2

When we multiply each of these by (h/2pi) we obtain the quantizations of L(z) as depicted in the diagram.

The angle (theta) can indeed be found using the cosine relation (adjacent over the hypoteneuse) which yields (using 't' for theta):

cos (t) = L(z)/ L = m_l/ [l (l+1)]^1/2

Of course, it ought to be self-evident that we are obtaining allowed values for the angle, since obviously m_l is going to range from -2 to +2)

Thus, for the problem we've considered (with l = 2) we have the allowed cosines and angles:

m_l = 0 so cos(t) = 0 so t = 90 deg

m_l = 1 so cos (t) = 1/[6]^1/2 so t = 65.9 deg

m_l = -1 so cos(t) = -1/[6]^1/2 so t (theta) = 114.1 deg

m_l = 2 so cos(t) = 2/[6]^1/2 so t = 35. 2 deg

m_1 = -2 so cos(t) = -2/[6]^1/2 = 144.7 deg

2) Find the possible values of L, L(z) and theta for an electron in the 3d state of hydrogen.

This is very simple! For the electron in the 3d state we have: n = 3, l = 2

So: because the configuration fits perfectly with that detailed in (1) all the answers are the same!

3) The orbital angular momentum of the Earth in its orbit around the Sun is:

About Me

Specialized in space physics and solar physics, developed first astronomy curriculum for Caribbean secondary schools, has written thirteen books - the most recent:Fundamentals of Solar Physics. Also: Modern Physics: Notes, Problems and Solutions;:'Beyond Atheism, Beyond God', Astronomy & Astrophysics: Notes, Problems and Solutions', 'Physics Notes for Advanced Level&#39, Mathematical Excursions in Brane Space, Selected Analyses in Solar Flare Plasma Dynamics; and 'A History of Caribbean Secondary School Astronomy'. It details the background to my development and implementation of the first ever astronomy curriculum for secondary schools in the Caribbean.